We have represented the axis of the terral vortex as continually passing
through the centre of gravity of the earth and moon. Now, by following
out the principles of the theory, we shall see that this cannot be the
case, except when the moon is in quadrature with the sun. To explain
this:
[Illustration: Fig. 10]
Let the curve passing through C represent a portion of the orbit of the
earth, and S the sun. From the principles laid down, the density of the
ethereal medium increases outward as the square roots of the distances
from the sun. Now, if we consider the circle whose centre is C to
represent the whole terral vortex, it must be that the medium composing
it varies also in density at different distances from the sun, and at
the same time is rotating round the centre. That half of the vortex
which is exterior to the orbit of the earth, being most dense, has
consequently most inertia, and if we conceive the centre of gravity of
the earth and moon to be in the orbit (as it must be) at C, there will
not be dynamical balance in the terral system, if the centre of the
vortex is also found at C. To preserve the equilibrium the centre of the
vortex will necessarily come nearer the sun, and thus be found between T
and C, T representing the earth, and [MOON] the moon, and C the centre of
gravity of the two bodies. If the moon is in opposition, the centre of
the vortex will fall between the centre of gravity and the centre of the
earth, and have the apparent effect of diminishing the mass of the moon.
If, on the other hand, the moon is in conjunction, the centre of the
vortex will fall between the centre of gravity and the moon, and have
the apparent effect of increasing the mass of the moon. If the moon is
in quadrature, the effect will be null. The coefficient of this
inequality is 90', and depends on the sun's distance from the moon. When
the moon is more than 90d from the sun, this correction is positive, and
when less than 90d from the sun, it is negative. If we call this second
correction C, and the moon's distance from her quadratures Q, we have
the value of C = +/-(90' x sin Q)/R.
[Illustration: Fig. 11]
This correction, however, does not affect the inclination of the axis of
the vortex, as will be understood by the subjoined figure. If the moon
is in opposition, the axis of the vortex will not pass through C, but
through C', and QQ' will be parallel to KK'. If the moon is in
conjunction, the axis will be still parallel to KK', as represented by
the dotted line qq'. The correction, therefore, for displacement, is
equal to the arc KQ or Kq, and the correct position of the vortex on the
surface of the earth at a given time will be at the points Q or q and Q'
or q', considering the earth as a sphere.
[Illustration: Fig. 12]
In the spherical triangle APV, P is the pole of the earth, V the pole of
the vortex, A the point of the earth's surface pierced by the radius
vector of the moon, AQ is the corrected arc, and PV is the obliquity of
the vortex. Now, as the axis of the vortex is parallel to the pole V,
and the earth's centre, and the line MA also passes through the earth's
centre, consequently AQV will all lie in the same great circle, and as
PV is known, and PA is equal to the complement of the moon's declination
at the time, and the right, ascensions of A and V give the angle P, we
have two sides and the included angle to find the rest, PQ being the
complement of the latitude sought.
We will now give an example of the application of these principles.
_Example._[10] Required the latitude of the central vortex at the time
of its meridian passage in longitude 88d 50' west, July 2d, 1853.